Parallel algorithms have focused an increased interest due to advantages in computation time and quality of solutions when applied to industrial engineering problems. This communication is a survey and classification of works in the field of hybrid algorithms implemented in parallel and applied to combinatorial optimization problems similar to the to the permutation flowshop problem with the objective of minimizing the makespan, Fm|prmu|Cmax according to the Graham notation, the travelling salesman problem (TSP), the quadratic assignment problem (QAP) and, in general, those whose solution can be expressed as a permutation

González Rodríguez, Pedro Luis

León Blanco, José Miguel

Molina Pariente, José Manuel

Pérez González, Paz

English

idUS. Depósito de Investigación Universidad de Sevilla

1075  3rd International Conference on Industrial Engineering and Industrial Management  XIII Congreso de Ingeniería de Organización Barcelona-Terrassa, September 2nd-4th 2009 A survey of parallel hybrid applications to the permutation flow shop scheduling problem and similar problems  Jose Miguel León Blanco1, Pedro Luis González Rodríguez1, Paz Pérez González1, Jose Manuel Molina Pariente1 1 Dpto. de Organización Industrial y Gestión de Empresas. Escuela Técnica Superior de Ingenieros de Sevilla. Universidad de Sevilla. Cº. Descubrimientos s/n, 410952. Sevilla. miguel@esi.us.es, pedroluis@esi.us.es, pazperez@esi.us.es, jmolinap@esi.us.es Keywords: Branch and bound, parallel algorithms, metaheuristics, permutation flow-shop, TSP 1. Introduction Parallel algorithms have focused an increased interest due to advantages in computation time and quality of solutions when applied to industrial engineering problems. This communication is a survey and classification of works in the field of hybrid algorithms implemented in parallel and applied to combinatorial optimization problems similar to the to the permutation flowshop problem with the objective of minimizing the makespan, Fm|prmu|Cmax according to the Graham notation, the travelling salesman problem (TSP), the quadratic assignment problem (QAP) and, in general, those whose solution can be expressed as a permutation. 2. Parallel hybrid algorithms It has been demonstrated that collaboration between metaheuristics leads to better quality solutions and more robust methods, Talbi (2002), Basseur et al. (2004). This kind of collaboration between metaheuristics is known as hybrid algorithms, see Bachelet et al. (1996). We will classify parallel hybrid algorithms according to the type of single algorithms combined: Algorithms that combine only metaheuristics and algorithms that combine exact and approximate methods. 2.1. Hybrid algorithms combining approximate methods In the field of hybrid algorithms, most work is devoted to this kind of parallel methods. There are several surveys in this field, including those of Talbi (2002), Cotta et al. (2005) and Raidl (2006). These works show a great collection of hybrid methods but most of them are sequential algorithms. We will expand here (table 1) the survey carried out by Loiola et al (2007), paying attention to parallel hybrid methods. (SA: Simulated annealing, GA: Genetic Algorithm, LS: Local Search, ACO: Ant Colony Optimization, TS: Tabu Search).                                                    This work stems from the participation of the authors in a research project funded by the Spanish Ministry of Science and Innovation, grant DPI2007-61345, title Advanced Systems for Integrated Order Management, and SCOPE, grant P08-TEP-3630, funded by the Andalusian Government. 1076  Methods Reference Problem SA+GA+LS Nearchou (2004) Fm|prmu|Cmax GA+SA Huntley and Brown (1991) QAP Talbi, Hafidi et al. (1998) QAP GA+LS Huntley and Brown (1996) QAP ACO+TS Talbi et al (2001) QAP GA+TS Matsumura et al. (2000) TSP Several Zobolas et al. (2008) Fm|prmu|Cmax Table 1. Parallel hybrid methods combining only approximate methods We would like to focus our attention in two works devoted to the Fm|prmu|Cmax problem. The first one is a complex one by Zobolas et al. (2008). It combines four constructive heuristics (NEH, Nawaz, et al. (1983), CDS, Campbell et al. (1970), Palmer (1965) and Gupta (1971)) with two metaheuristics (VNS, Hansen & Mladenovic (2001) and a genetic algorithm, Holland (1992)). It reaches high quality solutions even using a sequential design. Another sequential hybrid algorithm is the one presented in Nearchou (2004). The author adds features of genetic algorithms and local search to a simulated annealing algorithm. The results regarding the quality of solutions are of the same order of the best ones in literature applied to Taillard (1993) benchmark. We refer in (table 2) the quality of these solutions.   Problem  Jobs 20 20 20 50 50 50 100 100 100 200 200 500 Reference Machines 5 10 20 5 10 20 5 10 20 10 20 20 Yamada & Reeves (1997) Hybrid GA      -0.01   -0.24  0.04  Stutzle (1998) MMAS 0.41 0.59 0.41 0.15 2.19 2.48 0.20 0.93     Esquivel et al. (2002) MCMP–NEH 0.18 0.64 0.54          Esquivel et al. (2002) AC-NEH 0.21 1.17 0.86          Ying & Liao (2004) ACS 1.19 1.70 1.60 0.43 1.89 2.71 0.22 1.22 2.22 0.64 1.30 1.68Nearchou (2004) Hybrid SAA 0.14 0.18 0.14 0.06 0.33 1.78 0.17 0.33 2.11 0.41 2.04 1.41Table 2. ARPD of solutions obtained by hybrid sequential methods combining only approximate methods 2.1.1. Parallel algorithms Regarding parallel algorithms combining only approximate methods applied to permutation flowshop or similar problems, we resume them in (table 3), showing the main methods applied. (GA/EA: genetic algorithms/evolutionary algorithms). Matsumura et al. (2000) describe two types of parallel metaheuristics: The first one is called cPTS (Cooperative Tabu Search). In this case, solutions obtained via tabu search are recombined using concepts taken from simulated annealing using a so called EX (edge recombination crossover) operator. The innovation of this method is to select the processes that combine their solutions to obtain a new one using different methods. The first, takes from all processes, the one with the best solution. The second selects two random processes and pick the best solution from these two processes. 1077  Method\Method GA/EA SA TS Matsumura et al. (2000) Mack et al. (2004) LS Denzinger & Offerman (1999)  SA  Wodecki & Bozejko (2002) Table 3. Parallel hybrid methods combining only approximate methods The second one is called HPMH (Hybrid Parallel MetaHeuristic). This method employs only one process for each metaheuristic, one for tabu search and one for the genetic algorithm. The genetic algorithm starts the conversation, sending the best solution to tabu search process when a chromosome is dominating the whole population and prevents genetic algorithm to improve. Tabu search process replies sending the best so far solution as a starting point for a new population in the genetic algorithm process. Mack et al. (2004) combine simulated annealing with tabu search in parallel to solve the container loading problem. The solution to this problem can be presented as a packing sequence so is similar to the permutation flowshop. The authors begin with a sequential simulated annealing and observed an improvement in quality with respect to tabu search method. Later, they used a hybrid algorithm combining tabu search and simulated annealing, obtaining better quality solutions. The last step was to implement this hybrid algorithm in parallel, to improve even more the quality of solutions. 2.2. Hybrid algorithms combining exact and approximate methods There are very few references describing applications of hybrid algorithms combining exact and approximate methods. Those applied to permutation problems are only a few of them. We will begin citing some of the sequential algorithms in literature. The only exact method employed is branch and bound. In (table 4) we have collected some of the works that describe sequential hybrid algorithms combining metaheuristics and exact methods. We have included not only parallel applications but also distributed ones when applied to the Fm|prmu|Cmax and similar problems. Methods B&B Problem TS Glover et al. (1996) Woodcock & Wilson (2005) Fm|prmu|Cmax GA Nagar et al. (1996) French et al. (2001) Fm|prmu|Cmax LS Haouari & Ladhari (2003) Framinan & Pastor (2008) Fm|prmu|Cmax Table 4. Sequential hybrid methods combining approximate and exact methods Haouari & Ladhari (2003) insert a branch and bound algorithm inside a local search one to obtain solutions to the permutation flowshop problem. In the next table (table 5) we have compiled some results of hybrid methods applied to instances of well known Taillard’s (1993) benchmark. 1078   (nxm) 20 x 5 20 x 10 20 x 20 50 x 5 50 x 10 50 x 20 100 x 5 Instance ta001 ta011 ta021 ta031 ta041 ta051 ta061 TAI 1278 1582 2297     NS    2724 3025 3875 5493 LH 1278 1582  2724 2291  5493 t <1 45  <1 111  <1 BBLS 1278 1582  2724 2991  5493 t <1 67  <1 2994  <1 HPSO 1278 1582 2297 2724 3034 3923 5493 t 1,9 3,7 7,5 29,8 249,9 560,6 353,4 BDS 1278 1582 2297 2724 3035 3911 - t 225-900* - - - - - - NEGAVNS 1278 1582 2297 2724 3021 3874 5493 t 1 4 26 6 38 22 34 Table 5. Quality (makespan) and time (s) needed by several sequential hybrid algorithms applied to some of Taillard’s (1993) Fm|prmu|Cmax problem instances The results compared in (table 5) come from the following methods: TAI – Taillard (1993), NS – Nowicki & Smutnicki (1996), LH – Ladhari & Haouari (2005), BBLS –  Haouari & Ladhari (2003), HPSO – Kuo et al. (2008), BDS – Framinan & Pastor (2008) and NEGAVNS – Zobolas et al. (2008). Unfortunately there are instances that cannot be compared due to lack of experiments and in other hand, we don’t have references about the time to reach the best solution with BDS algorithm. Also, it has to be noticed that experiments have been carried out in different computers with different level of power so the time to reach the minimum cannot be easily compared.  2.2.5. 2.1.1. Parallel algorithms There are, as in sequential methods, very few references reporting the use of parallel hybrid methods that combine exact and approximate methods applied to the permutation flowshop or similar problems (table 5).   Exact Method B&B Approx. method GA/EA Pessan et al. (2008) Denzinger & Offerman (1999) Cotta et al. (1995) SA Nwana et al. (2005) Table 5. Parallel hybrid methods combining approximate and exact methods Pessan et al. (2008) combine branch and bound with a genetic algorithm and apply it to scheduling with restart problems. Some of the methods  We have found several references to the use of agents systems or teams of agents. These methods can employ a different algorithm in each of the agents and build a parallel hybrid method. Puchinger & Raidl (2005) refers this architecture. The architecture is known as teams of agents, Asynchronous Teams (A-Teams) as described by Talukdar et al. (1998), by Talukdar (1998) and by Talukdar (1992) or TEams for Cooperative Heterogeneous Search (TECHS) as presented by Denzinger & Offerman (1999). 1079  3. Future work We are working now in a parallel hybrid algorithm combining B&B and CLM, see Ghosh and Sierksma (2002), based in BDS (Bound Driven Search), see Framinan and Pastor (2008). This algorithm will employ some of the ideas found in this survey regarding the collaboration and communication between processes. References Bachelet,V.; Preux,P.; Talbi,Eg. (1996). 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A survey of parallel hybrid applications to the permutation flow shop scheduling problem and similar problems

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A survey of parallel hybrid applications to the permutation flow shop scheduling problem and similar problems

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